She wants a “backstop” measure which would create a temporary “joint customs territory” with the EU for the whole of the UK.

But Brussels appears set to insist on a Northern Ireland-only “backstop to the backstop” in case negotiations on a wider UK approach break down.

Meanwhile, a long-awaited Brexit deal could be brokered within a week, senior EU leader Donald Tusk has claimed.

Sparking fervent denials from Downing Street, the European Council President told Channel 4 News on Thursday there could be a breakthrough in negotiations in as little as “maybe five, maybe six, maybe seven days”.

In the letter to the DUP, May said: “I am clear that I could not accept there being any circumstances or conditions in which that ‘backstop to the backstop’, which would break up the UK customs territory, could come in to force.”

But the DUP has interpreted the wording of her letter to mean that the measure will be contained in the Brexit divorce deal despite Mrs May’s insistence it will never come into effect.

Brexiteer MPs have called on Theresa May to publish the full details of the backstop arrangement